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Kurt Gödel

Kurt Gödel
Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German: [ˈkʊʁt ˈɡøːdəl] ( ); April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[1] A. N. Whitehead,[1] and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. Life[edit] Childhood[edit] In his family, young Kurt was known as Herr Warum ("Mr. Studying in Vienna[edit]

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Gödel numbering A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects. Simplified overview[edit]

Turing completeness In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing complete or computationally universal if it can be used to simulate any single-taped Turing machine. The concept is named after Alan Turing. A classic example is lambda calculus. Computability theory includes the closely related concept of Turing equivalence. Two computers P and Q are called Turing equivalent if P can simulate Q and Q can simulate P. One Hundred Interesting Mathematical Calculations, Number 9: Archive Entry From Brad DeLong's Webjournal One Hundred Interesting Mathematical Calculations, Number 9 One Hundred Interesting Mathematical Calculations, Number 9: False Positives Suppose that we have a test for a disease that is 98% accurate: if one has the disease, the test comes back "yes" 98% of the time (and "no" 2% of the time), and if one does not have the disease, the test comes back "no" 98% of the time (and "yes" 2% of the time). Suppose further that 0.5% of people--one out of every two hundred--actually has the disease. Your test comes back "yes."

Bertrand Russell Russell led the British "revolt against idealism" in the early 20th century.[58] He is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and his protégé Ludwig Wittgenstein. He is widely held to be one of the 20th century's premier logicians.[55] With A. N. Prime Curios!: 9901 Turn this prime number upside down to get the year of the Norman Conquest of England. Any 12-digit number that has three or four digits that repeat themselves throughout the number will have 9901 as a prime factor. [Schuler] 9901, 99990001, 999999000001, and 9999999900000001, are primes. [Avrutin]

Gödel, Escher, Bach Gödel, Escher, Bach: An Eternal Golden Braid (pronounced [ˈɡøːdəl ˈɛʃɐ ˈbax]), also known as GEB, is a 1979 book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll".[1] By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. FIBONACCI "...considering both the originality and power of his methods, and the importance of his results, we are abundantly justified in ranking Leonardo of Pisa as the greatest genius in the field of number theory who appeared between the time of Diophantus [4th century A.D.] and that of Fermat" [17th century] R.B. McClenon [13]. [Numbers in square brackets refer to REFERENCES at the end of this article.] 1. The world of Fibonacci.

Rudolf Carnap Rudolf Carnap (May 18, 1891 – September 14, 1970) was a German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. Life and work[edit] Carnap's Birthplace in Wuppertal Orthogonality The line segments AB and CD are orthogonal to each other. The concept of orthogonality has been broadly generalized in mathematics, science, and engineering, especially since the beginning of the 16th century. Much of the generalizing has taken place in the areas of mathematical functions, calculus and linear algebra.

Schrödinger's cat Schrödinger's cat: a cat, a flask of poison, and a radioactive source are placed in a sealed box. If an internal monitor detects radioactivity (i.e. a single atom decaying), the flask is shattered, releasing the poison that kills the cat. The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead.

Eliminative materialism Eliminativists argue that modern belief in the existence of mental phenomena is analogous to the ancient belief in obsolete theories such as the geocentric model of the universe. Eliminativism stands in opposition to reductive materialism, which argues that a mental state is well defined, and that further research will result in a more detailed, but not different understanding.[3] An intermediate position is revisionary materialism, which will often argue that the mental state in question will prove to be somewhat reducible to physical phenomena - with some changes to the common sense concept. Eliminativism about a class of entities is the view that that class of entities does not exist.[4] For example, all forms of materialism are eliminativist about the soul; modern chemists are eliminativist about phlogiston; and modern physicists are eliminativist about the existence of luminiferous aether. Overview[edit] Philosophers who argue against eliminativism may take several approaches.